Discrete Optimization
Iterated tabu search for the circular open dimension problem

https://doi.org/10.1016/j.ejor.2012.10.022Get rights and content

Abstract

This paper investigates the circular open dimension problem (CODP), which consists of packing a set of circles of known radii into a strip of fixed width and unlimited length without overlapping. The objective is to minimize the length of the strip. In this paper, CODP is solved by a series of sub-problems, each corresponding to a fixed strip length. For each sub-problem, an iterated tabu search approach, named ITS, is proposed. ITS starts from a randomly generated solution and attempts to gain improvements by a tabu search procedure. After that, if the obtained solution is not feasible, a perturbation operator is subsequently employed to reconstruct the incumbent solution and an acceptance criterion is implemented to determine whether or not accept the perturbed solution. As a supplementary method, the length of the strip is determined in monotonously decreasing way, with the aid of some post-processing techniques. The search terminates and returns the best found solution after the allowed computation time has been elapsed. Computational experiments based on numerous well-known benchmark instances show that ITS produces quite competitive results, with respect to the best known results, while the computational time remains reasonable for each instance.

Highlights

► An effective iterated tabu search algorithm (ITS) is originally proposed for CODP. ► A robust tabu search procedure is implemented. ► A solution perturbation operator as well as an acceptance criterion is designed. ► ITS improves 76 best known results out of 146 instances within reasonable time.

Introduction

Cutting and packing (C&P) problems are widely encountered in practical applications, such as paper industry (Fraser and George, 1994), wireless communication (Adickes et al., 2002), marine transport (Birgin et al., 2005), aircraft designing (Liu and Li, 2010), and material cutting (He et al., 2012). They generally consist of cutting (or packing) a set of small items from (or into) a large object so as to minimize the wasted portion. As well-known NP-hard problems, C&P problems are extremely challenging to solve exactly and thus heuristics which attempt to obtain approximate solutions within reasonable time have been the most proposed approaches for tackling C&P problems.

As a representative variant of the C&P family, the circle packing problem (denoted by CPP) is concerned about how to pack a number of circles of known radii into a larger container without overlapping. The objective is to minimize the container size. According to the size of the circles to pack, CPP can be classified into two categories (Castillo et al., 2008): the arbitrary sized circle packing problem (denoted by ACP) and the uniform sized circle packing problem (denoted by UCP). It should be pointed out that ACP has to deal with both continuous and combinatorial features of the problem, while UCP mainly deals with continuous optimization problem. Due to the extremely large-scale combinatorial solution space of ACP, the approaches proposed for ACP are usually very different from those proposed for UCP.

This paper investigates a variant of ACP: the circular open dimension problem (denoted by CODP), which attempts to pack N(N = 1, 2, …) arbitrary sized circles into a strip of fixed width and unlimited length without overlapping. More precisely, CODP can be formulated as follows.

Given a strip of fixed width W and unlimited length L, as well as N arbitrary sized circles Ci of known radii ri(i = 1, 2,  , N). Take the origin of two-dimensional Cartesian coordinate system at the midpoint of the container, and denote the coordinates of the midpoint of Ci by (xi, yi). The objective of CODP is to obtain a solution (X, L), where X is a configuration denoted by (x1, y1,  , xi, yi,  , xN, yN), such that.

Minimize L, subject to:|xi|+ri0.5L1iN,|yi|+ri0.5W1iN,(xi-xj)2+(yi-yj)2ri+rj1i<jN.

The first two constraints state that each circle should not extend outside the container. The third constraint requires that any pair-wise circles cannot overlap with each other. (X, L) is termed a feasible solution if it meets all the constraints.

Furthermore, in order to measure the feasibility of a given solution (X, L), we define a penalty function based on the definition of overlaps as follows. For any solution (X, L), there may exist two kinds of overlaps: overlaps between two circles and overlaps between a circle and a border of the strip. Respectively, the overlapping depth between the ith circle Ci and the jth circle Cj isOij=Max0,ri+rj-(xi-xj)2+(yi-yj)2.

And the overlapping depth between Ci and a vertical border of the strip isOix=Max{0,ri+|xi|-0.5L}.

Similarly, the overlapping depth between Ci and a horizontal border of the strip isOiy=Max{0,ri+|yi|-0.5W}.

By adding all squares of overlapping depth together, we get a penalty function E(X, L) which measures the feasibility of a solution (X, L) as followsE(X,L)=i=1N-1j=i+1NOij2+i=1NOix2+Oiy2.

According to this definition, it is not difficult to find that: (1) For any solution (X, L), E(X, L)  0. (2) (X, L) is feasible if and only if E(X, L) = 0. Therefore, the objective of CODP is to minimize L while guaranteeing E(X, L) = 0.

In the present work, we try to solve CODP by a series of sub-problems, each with a fixed strip length. For each sub-problem, an iterated tabu search approach named ITS is proposed, which is composed of a tabu search procedure and a perturbation operator, associated with an acceptance criterion. As supplementary methods, some post-processing techniques are implemented to determine the strip length in monotonously decreasing mode. Computational results demonstrate that ITS is rather competitive with respect to the state-of-the-art approaches, in terms of both solution quality and computation time.

The rest of this paper is organized as follows: Section 2 briefly reviews the relevant literature. Section 3 presents the details of the proposed ITS algorithm. Computational experiments and analysis are presented in Sections 4 Computational results, 5 Conclusion concludes the paper.

Section snippets

Related literature

Various approaches have been developed for solving the circle packing problems (CPP). In this section, we briefly review the approaches proposed for solving CPP, especially for the circular open dimension problem (CODP) and its closely related variants.

As mentioned above, CPP can be classified into two categories: the arbitrary sized circle packing problem (ACP) and the uniform sized circle packing problem (UCP). Herein, we introduce several representative approaches proposed for ACP at first

Proposed approach

This paper studies CODP, which considers how to pack a number of arbitrary sized circles into a strip of fixed width and unlimited length without overlapping. In this paper, the original optimization problem is solved by a series of sub-problems, each considers about how to feasibly pack all the circles into a rectangular container of fixed width and fixed length. As supplementary techniques, we try to determine the strip length in a monotonously decreasing mode, just as detailed as follows.

  • (1)

Computational results

In order to evaluate the performance of the proposed approach, we implement ITS (as detailed in Algorithm 1) in C++ language and run it on several computers, each with an Intel Xeon E5440 2.83 gigahertz processor and 2 gigabytes RAM. Note that when doing the computational experiments, each benchmark instance occupies only one processor.

Two sets of 146 representative instances, which have been widely used as benchmarks by previous researchers, are taken from the literature and tested.

Conclusion

Cutting and packing (C&P) problems are well known NP-hard problems and are widely encountered in practical applications. This paper investigates the circular open dimension problem (CODP), which is a representative variant of the C&P family. For this problem, an iterated tabu search algorithm named ITS is proposed, which is composed of a tabu search procedure (TS) and a solution perturbation operator associated with an acceptance criterion. As a representative perturbation-based approach, the

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 61173180, 61100144, 61100076 and 71201119). Sincere thanks to the reviewers for their professional comments and suggestions which helped us to improve the quality of this paper significantly. Bo Peng also provided a lot of help.

References (43)

  • W.Q. Huang et al.

    Global optimization method for finding dense packings of equal circles in a circle

    European Journal of Operational Research

    (2011)
  • W.Q. Huang et al.

    New heuristics for packing unequal circles into a circular container

    Computers & Operations Research

    (2006)
  • T. James et al.

    A cooperative parallel tabu search algorithm for the quadratic assignment problem

    European Journal of Operational Research

    (2009)
  • J.F. Liu et al.

    An improved energy landscape paving algorithm for the problem of packing circles into a larger containing circle

    Computers and Industrial Engineering

    (2009)
  • Z.P. et al.

    Adaptive Tabu Search for course timetabling

    European Journal of Operational Research

    (2010)
  • C.D. Maranas et al.

    New results in the packing of equal circles in a square

    Discrete Mathematics

    (1995)
  • Y.G. Stoyan et al.

    Mathematical model and solution method of optimization problem of placement of rectangles and circles taking into account special constraints

    International Transactions in Operational Research

    (1998)
  • Y.G. Stoyan et al.

    A mathematical model and a solution method for the problem of placing various-sized circles into a strip

    European Journal of Operational Research

    (2004)
  • H.Q. Wang et al.

    An improved algorithm for the packing of unequal circles within a larger containing circle

    European Journal of Operational Research

    (2002)
  • G. Wäscher et al.

    An improved typology of cutting and packing problems

    European Journal of Operational Research

    (2007)
  • Q.H. Wu et al.

    Coloring large graphs based on independent set extraction

    Computers & Operations Research

    (2012)
  • Cited by (30)

    • Adaptive simulated annealing with greedy search for the circle bin packing problem

      2022, Computers and Operations Research
      Citation Excerpt :

      The quasi-physical methods are based on a physical gradient or human-intuitive behavior to enhance the solutions obtained by problem-oriented heuristics (Wang et al., 2002; Lubachevsky and Graham, 1997; He et al., 2018), while meta-heuristic optimizations usually have an evaluation function devised to employ a trade-off between randomization and local search, with the goal of directing and remodeling basic heuristics to generate feasible solutions. Typical algorithms include a simulated annealing approach (SA) (Hifi et al., 2004), monotonic basin hopping approach (MBH) (Grosso et al., 2010), iterated tabu search (ITS) (Fu et al., 2013), action-space-based global optimization algorithm (ASGO) (He et al., 2015), formulation space search (FSS) (López and Beasley, 2016), adaptive tabu search and variable neighborhood descent (ATS-VND) (Zhizhong et al., 2018), etc. Most of the constructive solutions focus on the traditional CPP and are designed on the specific characteristics of the problem.

    • Adaptive Tabu search and variable neighborhood descent for packing unequal circles into a square

      2018, Applied Soft Computing Journal
      Citation Excerpt :

      The supplementary neighborhoods are less efficient and more time-consuming than the basic neighborhood used in Best Improvement search, and are arranged in descending order according to their efficiencies. Both Tabu search and Variable Neighborhood search have provided outstanding results on all kinds of optimization problems [4–9]. Their outstanding effectiveness and generality have attracted some researchers to further adapt and combine them to provide even better results [5–8].

    • A memetic algorithm to pack unequal circles into a square

      2018, Computers and Operations Research
      Citation Excerpt :

      Different from the crossover operator, it generates an offspring solution basically from one parent solution rather than two. Previous perturbation operators in Iterated local search algorithms for unequal circles packing problems usually randomly reset or swap the randomly chosen circles (Zeng et al., 2016; Fu, 2011; Huang et al., 2012; Ye et al., 2013; Fu et al., 2013). There are two differences in our perturbation operator.

    View all citing articles on Scopus
    View full text